Western allies and Turkey have secured a deal to put the entire military campaign against Muammar Gaddafi under Nato command by next week, UK and French sources have told the Guardian.

The US, Britain, France and Turkey agreed to put the three-pronged offensive – a no-fly zone, an arms embargo, and air strikes – under a Nato command umbrella, in a climbdown by France that accommodates strong Turkish complaints about the scope and control of the campaign.

The deal appeared to end days of infighting among western allies, but needed to be blessed by all 28 Nato member states. At the end of a four-day meeting of Nato ambassadors in Brussels, Anders Fogh Rasmussen, the secretary general, said Nato had agreed to take command of the no-fly zone from the Americans. Disputes have raged at Nato HQ every day this week. Rasmussen contradicted leading western officials by announcing that Nato's authority was limited to commanding the no-fly zone, but he signalled there was more negotiation to come.

"At this moment, there will still be a coalition operation and a Nato operation," he said. This meant Nato would command the no-fly zone and police the arms embargo. But on the most contentious part, air strikes and ground attacks against Gaddafi, consensus remained elusive.

The agreement emerged from phone calls between William Hague, the foreign secretary, Alain Juppé, the French foreign minister, Hillary Clinton, the US secretary of state, and Ahmet Davutoglu, the Turkish foreign minister, following rancorous attacks from the Turkish leadership on French ambitions to lead the anti-Gaddafi war effort.

The agreement also gives political oversight of the military action to a committee of the international coalition in the campaign. Since the no-fly zone and air attacks on Libya began last Saturday by France, Washington has been in charge of operations, but is eager to surrender the role.

Under the scheme agreed, the transfer to Nato will take place by the latest in London on Tuesday, when the parties to the coalition against Gaddafi gather in London for a special "contact group" conference. French sources said the Benghazi-based Libyan rebel leadership would be in London to attend. The conference will consist of two meetings: a war council made up of the main governments taking part in the military action, as well as a broader assembly including Arab and African countries devoted to Libya's future.

Hillary Clinton welcomed the Nato decision to take command of the Libyan operations and police the no-fly zone, and she expected that it would eventually take over responsibility for protecting civilians, enforcing an arms embargo and supporting the humanitarian mission. "We are taking the next step. We have agreed along with our Nato allies to transition command and control for the no-fly zone over Libya to Nato. All 28 allies have also now authorised military authorities to develop an operations plan for Nato to take on the broader civilian protection mission," she said.

She said the United Arab Emirates was to join Qatar in sending planes to enforce the no-fly zone.

Barack Obama, who returned to Washington on Wednesday, is reluctant to make a televised address to the nation about Libya because he is keen to try to keep it low-key. Administration officials, as part of this strategy, pointedly refuse to call it a war.

Republicans have been calling on him to explain the mission. The president has also faced criticism from his own Democratic party.

"I think he needs to face the nation and tell the nation, and tell Congress, what the end game is and how this going to play out," Senator Sherrod Brown, a Democrat, said on MSNBC.President Nicolas Sarkozy, who had tried to diminish the role of Nato, conceded, in the face of Turkish opposition, that a two-tier structure would run the operation: Nato "assets" will co-ordinate all aspects, including enforcement of the no-fly zone, protecting civilians through air strikes, and enforcing a UN arms embargo. Juppé agreed that Nato would be in control of the entire operation.

Political oversight will be in the hands of a committee of a smaller number of countries involved in the military campaign.

There had been bitter attacks from the Turkish government on Sarkozy's leadership of the campaign, accusing the French of lacking a conscience in their conduct of operations, with criticism from the prime minister, Recep Tayyip Erdogan, and the president, Abdullah Gül.

France had insisted on Tuesday that the operations would be "non-Nato". Turkey was emphatically behind sole Nato control of the operations. In Istanbul, Erdogan said: "I wish that those who only see oil, gold mines and underground treasures when they look in [Libya's] direction, would see the region through glasses of conscience from now on."

This week, Claude Guéant, the French interior minister who was previously Sarkozy's chief adviser, angered the Muslim world by stating that the French president was "leading a crusade" to stop Gaddafi massacring Libyans. Russian prime minister Vladimir Putin also used the word in reference to air strikes on Libya. And George Bush had notoriously used the word after the 11 September 2001 attacks on the US that led to the Iraq war.

Erdogan said: "Those who use such hair-raising, frightening terms that fuel clashes of civilisations, or those who even think of them, need to immediately evaluate their own conscience."

The Turks are incensed at repeated snubs by Sarkozy. The French failed to invite Turkey to last Saturday's summit in Paris, which preceded the air strikes. French fighters taking off from Corsica struck the first blows. The Turkish government accused Sarkozy of launching not only the no-fly zone, but his presidential re-election campaign.

The dispute over Libya appears highly personal. Sarkozy went to Turkey last month for the first time in four years as president, but the visit was repeatedly delayed and then downgraded from a state presidential event. He stayed in Turkey for five hours. "Relations between Turkey and France deserve more than this," complained Erdogan. "I will speak with frankness. We wish to host him as president of France. But he is coming as president of the G20, not as that of France."